What you are describing should work without issue.
Your connection string should look like:
<add name="IntegratedAuthConnectionString" connectionString=
"Data Source=DATABASEINSTANCE;Integrated Security=True"
providerName="System.Data.SqlClient"/>
Your web.config should also specify windows authentication
<authentication mode="Windows"/>
Create a test page which dumps everything from the Request.ServerVariables collection and ensure that you are seeing the authenticated username and domain. Something along the lines of the code below
foreach (string s in Request.ServerVariables)
{
Response.Write(s + ": " + Request.ServerVariables[s] + "<br />");
}